{"paper":{"title":"Counting Polynomials with Distinct Zeros in Finite Fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Haiyan Zhou, Li-Ping Wang, Weiqiong Wang","submitted_at":"2017-02-08T08:50:46Z","abstract_excerpt":"Let $\\mathbb{F}_q$ be a finite field with $q=p^e$ elements, where $p$ is a prime and $e\\geq 1$ is an integer. Let $\\ell<n$ be two positive integers. Fix a monic polynomial $u(x)=x^n +u_{n-1}x^{n-1}+\\cdots +u_{\\ell+1}x^{\\ell+1} \\in \\mathbb{F}_q[x]$ of degree $n$ and consider all degree $n$ monic polynomials of the form $$f(x) = u(x) + v_\\ell(x), \\ v_\\ell(x)=a_\\ell x^\\ell+a_{\\ell-1}x^{\\ell-1}+\\cdots+a_1x+a_0\\in \\mathbb{F}_q[x].$$ For integer $0\\leq k \\leq {\\rm min}\\{n,q\\}$, let $N_k(u(x),\\ell)$ denote the total number of $v_\\ell(x)$ such that $u(x)+v_\\ell(x)$ has exactly $k$ distinct roots in $\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.02327","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}